Because of the numerous books that have already appeared about the classical Analysis, in principle it is very difficult to bring new facts in this field. However, the engineers, researchers in experimental sciences, and even the students actually need a quick and clear presentation of the basic theory, together with an extensive and efficient guidance to solve practical problems. Therefore, in this book we tried to combine the essential (but rigorous) theoretical results with a large scale of concrete applications of the Mathematical Analysis, and formulate them in nowadays language. The content is based on a two-semester course that has been given in English to students in Computer Sciences at the University of Craiova, during a couple of years. As an independent work, it contains much more than the effective lessons can treat according to the imposed program. Starting with the idea that nobody (even student) has enough time to read several books in order to rediscover the essence of a mathematical theory and its practical use, we have formulated the following objectives for the present book: 1. Accessible connection with mathematics in lyceum 2. Self-contained, but well referred to other works 3. Prominence of the specific structures 4. Emphasis on the essential topics 5. Relevance of the sphere of applications. The first objective is assured by a large introductory chapter, and by the former paragraphs in the other chapters, where we recall the previous notions. To help intuition, we have inserted a lot of figures and schemes. The second one is realized by a complete and rigorous argumentation of the discussed facts. The reader interested in enlarging and continuing the study is still advised to consult the attached bibliography. Besides classical books, we have mentioned the treatises most available in our zone, i.e. written by Romanian authors, in particular from Craiova. Because Mathematical Analysis expresses in a more concrete form the philosophical point of view that assumes the continuous nature of the Universe, it is very significant to reveal its fundamental structures, i.e. the topologies. The emphasis on the structures is especially useful now, since the discrete techniques (e.g. digital) play an increasing role in solving practical problems. Besides the deeper understanding of the specific features, the higher level of generalization is necessary for a rigorous treatment of the fundamental topics like continuity, differentiability, etc. To touch the fourth objective, we have organized the matter such that each chapter debates one of the basic aspects, more exactly continuity,VIII convergence and differentiability in volume one, and different types of integrals in part two. We have explained the utility of each topic by plenty of historic arguments and carefully selected problems. Finally, we tried to realize the last objective by lists of problems at the end of each paragraph. These problems are followed by answers, hints, and sometimes by complete solutions. In order to help the non-native speakers of English in talking about the matter, we recommend books on English mathematical terms, including pronunciation and stress, e.g. the Guide to Mathematical Terms [BT4]. Our experience has shown that most language difficulties concern speaking, rather than understanding a written text. Therefore we encourage the reader to insist on the phonetics of the mathematical terms, which is essential in a fluent dialog with foreign specialists. In spite of the opinion that in old subjects like Mathematical Analysis everything is done, we still have tried to make our book distinguishable from other works. With this purpose we have pointed to those research topics where we have had some contributions, e.g. the quasi-uniform convergence in function spaces (§ II.3 in connection to [PM2] and [PM3]), the structures of discreteness (§ III.2 with reference to [BT3]), the unified view on convergence and continuity via the intrinsic topology of a directed set, etc. We also hope that a note of originality there results from: The way of solving the most concrete problems by using modern techniques (e.g. local extrema, scalar and vector fields, etc.); A rigorous but moderately extended presentation of several facts (e.g. higher order differential, Jordan measure in R n , changing the variables in multiple integrals, etc.) which sometimes are either too much simplified in practice, or too detailed in theoretical treatises; The unitary treatment of the Real and Complex Analysis, centered on the analytic (computational) method of studying functions and their practical use (e.g. § II.4, § IV.5, Chapter X, etc.). We express our gratitude to all our colleagues who have contributed to a better form of this work. The authors are waiting for further suggestions of improvements, which will be welcome any time. The Authors
Maria Predoi Trandafir Bălan MATHEMATICAL ANALYSIS VOL. I DIFFERENTIAL CALCULUS Craiova, 2005 VII PREFACE Because of the numerous books that have already appeared about the classical Analysis, in principle it is very difficult to bring new facts in this field. However, the engineers, researchers in experimental sciences, and even the students actually need a quick and clear presentation of the basic theory, together with an extensive and efficient guidance to solve practical problems. Therefore, in this book we tried to combine the essential (but rigorous) theoretical results with a large scale of concrete applications of the Mathematical Analysis, and formulate them in nowadays language. The content is based on a two-semester course that has been given in English to students in Computer Sciences at the University of Craiova, during a couple of years. As an independent work, it contains much more than the effective lessons can treat according to the imposed program. Starting with the idea that nobody (even student) has enough time to read several books in order to rediscover the essence of a mathematical theory and its practical use, we have formulated the following objectives for the present book: 1. Accessible connection with mathematics in lyceum 2. Self-contained, but well referred to other works 3. Prominence of the specific structures 4. Emphasis on the essential topics 5. Relevance of the sphere of applications. The first objective is assured by a large introductory chapter, and by the former paragraphs in the other chapters, where we recall the previous notions. To help intuition, we have inserted a lot of figures and schemes. The second one is realized by a complete and rigorous argumentation of the discussed facts. The reader interested in enlarging and continuing the study is still advised to consult the attached bibliography. Besides classical books, we have mentioned the treatises most available in our zone, i.e. written by Romanian authors, in particular from Craiova. Because Mathematical Analysis expresses in a more concrete form the philosophical point of view that assumes the continuous nature of the Universe, it is very significant to reveal its fundamental structures, i.e. the topologies. The emphasis on the structures is especially useful now, since the discrete techniques (e.g. digital) play an increasing role in solving practical problems. Besides the deeper understanding of the specific features, the higher level of generalization is necessary for a rigorous treatment of the fundamental topics like continuity, differentiability, etc. To touch the fourth objective, we have organized the matter such that each chapter debates one of the basic aspects, more exactly continuity, VIII convergence and differentiability in volume one, and different types of integrals in part two. We have explained the utility of each topic by plenty of historic arguments and carefully selected problems. Finally, we tried to realize the last objective by lists of problems at the end of each paragraph. These problems are followed by answers, hints, and sometimes by complete solutions. In order to help the non-native speakers of English in talking about the matter, we recommend books on English mathematical terms, including pronunciation and stress, e.g. the Guide to Mathematical Terms [BT 4 ]. Our experience has shown that most language difficulties concern speaking, rather than understanding a written text. Therefore we encourage the reader to insist on the phonetics of the mathematical terms, which is essential in a fluent dialog with foreign specialists. In spite of the opinion that in old subjects like Mathematical Analysis everything is done, we still have tried to make our book distinguishable from other works. With this purpose we have pointed to those research topics where we have had some contributions, e.g. the quasi-uniform convergence in function spaces (§ II.3 in connection to [PM 2 ] and [PM 3 ]), the structures of discreteness (§ III.2 with reference to [BT 3 ]), the unified view on convergence and continuity via the intrinsic topology of a directed set, etc. We also hope that a note of originality there results from: The way of solving the most concrete problems by using modern techniques (e.g. local extrema, scalar and vector fields, etc.); A rigorous but moderately extended presentation of several facts (e.g. higher order differential, Jordan measure in R n , changing the variables in multiple integrals, etc.) which sometimes are either too much simplified in practice, or too detailed in theoretical treatises; The unitary treatment of the Real and Complex Analysis, centered on the analytic (computational) method of studying functions and their practical use (e.g. § II.4, § IV.5, Chapter X, etc.). We express our gratitude to all our colleagues who have contributed to a better form of this work. The authors are waiting for further suggestions of improvements, which will be welcome any time. The Authors Craiova, September 2005 IX CONTENTS VOL. I. DIFFERENTIAL CALCULUS PREFACE VII Chapter I. PRELIMINARIES § I.1 Sets. Relations. Functions 1 Problems § I.1. 11 § I.2 Numbers 15 Problems § I.2. 26 § I.3 Elements of linear algebra 29 Problems § I.3. 43 § I.4 Elements of Topology 47 Part 1. General topological structures 47 Part 2. Scalar products, norms and metrics 52 Problems § I.4. 58 Chapter II. CONVERGENCE § II.1 Nets 61 Part 1. General properties of nets 61 Part 2. Sequences in metric spaces 65 Problems § II.1. 74 § II.2 Series of real and complex numbers 77 Problems § II.2. 89 § II.3 Sequences and series of functions 93 Problems § II.3. 111 § II.4 Power series 117 Problems § II.4. 130 X Chapter III. CONTINUITY § III.1 Limits and continuity in R 135 Problems § III.1. 142 § III.2 Limits and continuity in topological spaces 144 Problems § III.2. 155 § III.3 Limits and continuity in metric spaces 156 Problems § III.3. 163 § III.4 Continuous linear operators 165 Problems § III.4. 179 Chapter IV. DIFFERENTIABILITY § IV.1 Real functions of a real variable 183 Problems § IV.1. 187 § IV.2 Functions between normed spaces 188 Problems § IV.2. 200 § IV.3 Functions of several real variables 201 Problems § IV.3. 219 § IV.4 Implicit functions 224 Problems § IV.4. 241 § IV.5 Complex functions 245 Problems § IV.5. 260 INDEX 264 BIBLIOGRAPHY 270 1 CHAPTER I. PRELIMINARIES § I.1. SETS, RELATIONS, FUNCTIONS From the very beginning, we mention that a general knowledge of set theory is assumed. In order to avoid the contradictions, which can occur in such a “naive” theory, these sets will be considered parts of a total set T, i.e. elements of P (T). The sets are usually depicted by some specific properties of the component elements, but we shall take care that instead of sets of sets it is advisable to speak of families of sets (see [RM], [SO], etc). When operate with sets we basically need one unary operation A {A = {x T : x A} (complement), two binary operations (A, B) A B = {x T : x A or x B} (union), (A, B) A B = {x T : x A and x B} (intersection), and a binary relation A=B x A iff x B (equality). 1.1. Proposition. If A, B, C P (T), then: (i) A (B C)=(A B) C ; A (B C)=(A B) C (associativity) (ii) A (B C)=(A B) (A C); A (B C)=(A B) (A C) (distributivity) (iii) A (A B)=A ; A (A B)=A (absorption) (iv) (A {A) B=B ; (A {A) B=B (complementary) (v) A B=B A; A B=B A (commutativity). 1.2. Remark. From the above properties (i)-(v) we can derive the whole set theory. In particular, the associativity is useful to define intersections and unions of a finite number of sets, while the extension of these operations to arbitrary families is defined by }:{}:{ i i AxIiTxIiA and }:{}:{ i i AxthatsuchIiTxIiA . Some additional notations are frequent, e.g. = A {A for the (unique!) void set, A\B = A {B for the difference, A B = (A\B) (B\A) for the symmetric difference, A B (defined by A B = B) for the relation of inclusion, etc. Chapter I. Preliminaries 2 More generally, a non-void set A on which the equality = , and the operations , and (instead of {, , respectively ) are defined, such that conditions (i)-(v) hold as axioms, represents a Boolean algebra. Besides P (T), we mention the following important examples of Boolean algebras: the algebra of propositions in the formal logic, the algebra of switch nets, the algebra of logical circuits, and the field of events in a random process. The obvious analogy between these algebras is based on the correspondence of the following facts: - a set may contain some given point or not; - a proposition may be true or false; - an event may happen in an experience or not; - a switch may let the current flow through or break it; - at any point of a logical circuit may be a signal or not. In addition, the specific operations of a Boolean algebra allow the following concrete representations in switch networks: Similarly, in logical circuits we speak of “logical gates” like A A A e e A (double switch) A B A _ B (parallel connection) A B A B ^ (serial connection) A A e e ( -gate) non A A B B C C _ _ _ ( -gate) or A A B B C C ^ ^ ^ ( -gate) and § I.1. Sets, Relations, Functions 3 1.3. The Fundamental Problems concerning a practical realization of a switch network, logic circuits, etc., are the analysis and the synthesis. In the first case, we have some physical realization and we want to know how it works, while in the second case, we desire a specific functioning and we are looking for a concrete device that should work like this. Both problems involve the so-called working functions, which describe the functioning of the circuits in terms of values of a given formula, as in the table from bellow. It is advisable to start by putting the values 1, 0, 1, 0,… for A, then 1, 1, 0, 0,… for B, etc., under these variables, then continue by the resulting values under the involved connectors , , , etc. by respecting the order of operations, which is specified by brackets. The last completed column, which also gives the name of the formula, contains the “truth values” of the considered formula. As for example, let us consider the following disjunction, whose truth- values are in column (9): (A B) [( A C) + B] 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 1 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 (1)(6)(3) (9) (2) (7)(5) (8)(4) where (1), (2), etc. show the order of completing the columns. The converse problem, namely that of writing a formula with previously given values, makes use of some standard expressions, which equal 1 only once (called fundamental conjunctions). For example, if a circuit should function according to the table from below, A B C f(A,B,C) fundamental conjunctions 1 1 1 1 A B C 0 1 1 0 - 1 0 1 0 - 0 0 1 1 A B C 1 1 0 0 - 0 1 0 1 A B C 1 0 0 1 A B C 0 0 0 0 - Chapter I. Preliminaries 4 then one working function is f(A,B,C) = (A B C) (A B C) ( A B C) (A B C). This form of f is called normal disjunctive (see [ME], etc.). The following type of subfamilies of P (T), where T , is frequently met in the Mathematical Analysis (see [BN 1 ], [DJ], [CI], [L-P], etc.): 1.4. Definition. A nonvoid family F P (T) is called (proper) filter if [F 0 ] F ; [F 1 ] A, B F A B F ; [F 2 ] (A F and B A) B F . Sometimes condition [F 0 ] is omitted, and we speak of filters in generalized (improper) sense. In this case, F = P (T) is accepted as improper filter. If family F is a filter, then any subfamily B F for which [BF] A F B B such that B A, (in particular F itself) is called base of the filter F. 1.5. Examples. a) If at any fixed x R we define F P (T) by F = {A R: > 0 such that A (x – , x + )}, then F is a filter, and a base of F is B = { (x – , x + ): >0}. It is easy to see that {A R : A F } = {x}. b) The family F P (N), defined by F = {A N: n N such that A (n, )}, is a filter in P (N) for which B = {(n, ): n N} is a base, and {A N : A F } = . c) Let B P (R 2 ) be the family of interior parts of arbitrary regular polygons centered at some fixed (x, y) R 2 . Then F = {A R 2 : B B such that A B} is a filter for which family C , of all interior parts of the disks centered at (x, y), is a base (as well as B itself). 1.6. Proposition. In an arbitrary total set T we have: (i) Any base B of a filter F P (T) satisfies the condition [FB] A, B B C B such that C A B. (ii) If B P (T) satisfies condition [FB] (i.e. together with [F 0 ] it is a proper filter base), then the family of oversets G = {A T : B B such that A B} is a filter in P (T); we say that filter G is generated by B. (iii) If B is a base of F, then B generates F . § I.1. Sets, Relations, Functions 5 The proof is direct, and we recommend it as an exercise. 1.7. Definition. If A and B are nonvoid sets, their Cartesian product is defined by A B = {(a, b): a A, b B}. Any part R A B is called binary relation between A and B. In particular, if R T T, it is named binary relation on T. For example, the equality on T is represented by the diagonal = {(x, x): x T}. If R is a relation on T, its inverse is defined by R –1 = {(x, y): (y, x) R }. The composition of two relations R and S on T is noted R S = {(x, y): z T such that (x, z) S and (z, y) R }. The section (cut) of R at x is defined by R[x] = {y T : (x, y) R }. Most frequently, a binary relation R on T may be: Reflexive: R ; Symmetric: R = R –1 ; Antisymmetric: R R –1 = ; Transitive: R R R ; Directed: R [x] R [y] for any x, y T. The reflexive, symmetric and transitive relations are called equivalences, and usually they are denoted by . If is an equivalence on T , then each x T generates a class of equivalence, noted x^ = {y T : xy}. The set of all equivalence classes is called quotient set, and it is noted T/. The reflexive and transitive relations are named preorders. Any antisymmetric preorder is said to be a partial order, and usually it is denoted by . We say that an order on T is total (or, equivalently, (T, ) is totally, linearly ordered) iff for any two x, y T we have either x y or y x. Finally, (T, ) is said to be well ordered (or is a well ordering on T ) iff is total and any nonvoid part of T has a smallest element. 1.8. Examples. (i) Equivalences: 1. The equality (of sets, numbers, figures, etc.); 2. {((a, b) ,(c, d)) N 2 N 2 : a + d = b + c }; 3. {((a, b) ,(c, d)) Z 2 Z 2 : ad = bc }; [...]... ({B) = { [f (B)] holds for any B Y , while f ({A) and {[f (A)] generally cannot be compared The proof is left to the reader The following particular type of functions is frequently used in the Mathematical Analysis: 1.15 Definition Let S be a nonvoid set Any function f : N S is called sequence in S Alternatively we note f(n) = xn at any nN, and we mark the sequence f by mentioning the generic term... (P, Q), we practically reduce the study of complex functions of a complex variable to that of real vector functions of two real variables On this way, many problems of complex analysis can be reformulated and solved in real analysis This method will be intensively used in §III.4 (see also [HD], [CG], etc.) Alternatively, if z, the argument of f , is expressed in trigonometric form, then the image through... construction is possible using neighborhoods U, V, … of a fixed point x0 in any topological space The partitions, which occur in the definition of some integrals, generate directed sets (see the integral calculus) In particular, in order for us to define the Riemannian integral on [a, b] R, we consider partitions of the closed interval [a, b], i.e finite sets of subintervals of the form = { [ xk 1... is a member of the family iff each finite subset of A is) has a maximal member (Tukey) 1.20 Remark The axiom of choice will be adopted throughout this book, as customarily in the treatises on Classical Analysis Without insisting on each particular appearance during the development of the theory, we mention that the axiom of choice is essential in plenty of problems as for 10 § I.1 Sets, Relations, Functions... exactly, every nonvoid upper bounded subset of R has a supremum, which is known as Cantor’s axiom) 2.15 Remark Taking the Cantor’s axiom as a starting point of our study clearly shows that the entire Real Analysis is essentially based on the order completeness of R At the beginning, this fact is visible in the limiting process involving sequences in R (i.e in convergence theory), and later it is extended... (1,0) The list of systems of numbers can be continued; in particular, the spaces of dimension 2n can be organized as Clifford Algebra (see [C-E], etc.) 22 § I.2 Numbers Besides numbers, there are other mathematical entities, called vectors, tensors, spinors, etc., which can adequately describe the different quantities that appear in practice (see [B-S-T], etc.) Further refinements of the present classification... especially to underline that some properties are valid in both real and complex structures (e.g see the real and the complex linear spaces in §I.3, etc.) A special attention will be paid to the complex analysis, which turns out to be the natural extension and even explanation of many results involving real variables Step by step, the notion of real function of a real variable is extended to that of complex... (i) Sometimes we must extend the above notion of function, and allow that f(x) consists of more points; in such case we say that f is a multivalued (or one to many) function For example, in the complex analysis, f n is supposed to be an already known 1:n function Similarly, we speak of many to one, or many to many functions 8 § I.1 Sets, Relations, Functions This process of extending the action of... problem 11 from above), so that z 2 1 0 has no solutions, while z 2 1 0 has 4 solutions in this space 28 § I.3 ELEMENTS OF LINEAR ALGEBRA The linear structures represent the background of the Analysis, whose main purpose is to develop methods for solving problems by a local reduction to their linear approximations Therefore, in this paragraph we summarize some results from the linear algebra,